3D object representation 3D object representation are often divided into two broad categories, although not all representations fall neatly into one or the other of these two categories.
Boundary representations (B-reps)
Space – partitioning representations

Boundary representations (B-reps) :

Boundary representations (B-reps) Describe a three-dimensional object as a set of surfaces that separate the object interior from the environment.
Typical examples of boundary representations are polygon facets and spline patches

Space-partitioning representations :

Space-partitioning representations used to describe interior properties, by partitioning the spatial region containing an object into a set of small, non overlapping, contiguous solids (usually cubes).
A common space-partitioning description for a three-dimensional object is an octree representation.

Polygon surfaces :

Polygon surfaces The most commonly used boundary representation for a three-dimensional graphics object is a set of surface polygons that enclose the object interior.
Many graphics systems store all object descriptions as sets of surface polygons.
This simplifies and speeds up the surface rendering and display of objects, since all surfaces are described with linear equations.

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We define a polygon surface with a set of vertex coordinates and associated attribute parameters.
As information for each polygon is input, the data are placed into tables that are to be used in the subsequent processing, display, and manipulation of the objects in a scene.
Polygon data tables can be organized into two groups: geometric tables and attribute tables.
Geometric data tables contain vertex coordinates and parameters to identify the spatial orientation of the polygon surfaces.
Attribute information for an object includes parameters specifying the degree of transparency of the object and its surface reflectivity and texture characteristics.

Polygon representation :

Polygon representation Polygon can be represented by using polygon table.
A polygon table consists of vertex table, edge table and polygon surface table.

Vertex table :

Vertex table Co-ordinate values for each vertex in the object is stored in the vertex table.
Edge table
It contains points back into the vertex table to identify the vertices for each polygon edge.

Polygon surface table :

Polygon surface table It contains points back into the edge table to identify the edges for each polygon surface.

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X1,y1,z1 X3,y3,z3 X2,y2,z2 X4,y4,z4 X5,y5,z5

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Plane Surface Equation
The equation for a plane surface can be expressed as
Ax+By+Cz+D=0
Where (x, y, z) is any point on the plane , and the coefficients A, B,C, D are constants describing the spatial properties of the plane.

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We can obtain the values of A, B, C,D by solving a set of three plane equations using the coordinate
values for three non collinear points in the plane.
For this purpose, we can select three successive polygon vertices(x1,y1,z1), (x2,y2,z2), (x3,y3,z3) and solve the following set of simultaneous linear plane equations for the ratios A/D, B/D , and C/D

Polygon meshes :

Polygon meshes One type of a polygon mesh is a triangle strip.
This function produces n - 2 connected triangles, as shown in Fig.
Another similar function is the quadrilateral mesh, which generates a mesh of (n - 1) by (m - 1) quadrilaterals.

Polygon meshes :

Polygon meshes

QUADRIC SURFACES :

QUADRIC SURFACES A frequently used class of objects are the quadric surfaces, which are described with second-degree equations (quadratics).
They include spheres, ellipsoids, paraboloids, and hyperboloids.
Quadric surfaces, particularly spheres and ellipsoids, are common elements of graphics scenes, and they are often available in graphics packages as primitives from which more complex objects can be constructed.

sphere :

sphere

Ellipsoid :

Ellipsoid

Superquadrics :

Superquadrics Super quadrics are formed by incorporating additional parameters into the quadric equations to provide increased flexibility for adjusting object shapes.
The no: of additional parameters used is equal to the dimension of the object: one parameter for the curves and two parameters for surfaces.

superellipse :

superellipse The Cartesian representation for a superellipse is obtained from the equation of ellipse by allowing the exponents of x and y terms to be variable.
The Cartesian representation of superellipse equation is

Super ellipsoid :

Super ellipsoid

BLOBBY OBJECTS :

BLOBBY OBJECTS Some objects do not maintain a fixed shape, but change their surface characteristics in certain motions.
Examples in this class of objects include molecular structures, water droplets and other liquid effects, melting objects, and muscle shapes in the human body.
These objects can be described as exhibiting “blobbiness” and are often simply referred to as blobby objects, since their shapes show a certain degree of fluidity.

SPLINE :

SPLINE A spline is a flexible strip used to produce a smooth curve through a designated set of points.
Several small weights are distributed along the length of the strip to hold it in position as the curve is drawn
Spline curve in computer graphics refers to composite curve formed with polynomial sections satisfying specified continuity conditions at the boundary of the pieces.

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Spline Curve (simple) A set of 5 control points interpolated with piecewise continuous polynomial sections

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Spline Curve (Composite) A set of control points approximated with piecewise continuous polynomial sections.

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Control points are set of coordinate positions which indicates the shape of the curve.
When the piecewise curves are attached together
If the curve passes through each control point the resulting curve is said to interpolate the set of control points.
If the curve does not passes through each control point the resulting curve is said to approximate the set of control points
A spline curve is defined, modified and manipulated with operations on the control points.

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The spline curve can also be translated, rotated or scaled with transformations applied to the control points.
The convex polygon boundary that encloses a set of control points is called the convex hull.
The set of connected line segments is offered to as the control graph.

Bézier Curves & Surfaces :

Bézier Curves & Surfaces Named after the French engineer Pierre Bézier.
Bezier Splines are highly useful and convenient for curve and surface design.
Easy to implement hence widely used in CAD systems.
Bézier curves can be fitted to any no: of control points (at least three points to define a curve.)

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The main property of Bézier curve is that it always passes through the first and last control points (i.e. the boundary conditions at the two ends)
The two endpoints of the curve are called anchor points.
The other points, which define the shape of the curve, are called handles, tangent points, or nodes.
Attached to each handle are two control points. By moving the handles themselves, or the control points, you can modify the shape of the curve.

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Bézier Curves Bézier Curve

Hint: As a rule :

Hint: As a rule A Bézier curve is a polynomial of degree one less than the no: of control points used
Eg: 3 ctrl points – parabola
4 ctrl points – cubic curve

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Bezier curve can be specified with boundary condition or with blending functions Pk Pk+1 P(u) Pk-1 Pk+2 Cardinal spline section between control points Pk and Pk+1 P(u) determines the path of the Bezier curve between the control points P0 & Pn .i.e Pk and pk+1

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In general, given n+1 points, P0, P1, P2, …, Pn , the path of an approximating Bézier polynomial between P0 and Pn is P(u)=P0.Bezn,0(u)+P1.Bezn,1(u)+P2.Bezn,2(u)+…+Pn.Bezn,n(u)
where 0 u 1
i.e P(u) = Pk.Bezk,n(u), 0 u 1
P(u) – position vector
Pk=(xk,yk,zk) where k=0 to 1
n= number of control points
Bezk,n(u) – Bezier blending function Bernstein polynomials K=0 n Binomial coefficients

Slide 48:

Set of parametric equations for the individual curve coordinates are
x(u) = xk Bezk,n (u)
y(u) = yk Bezk,n (u)
z(u) = zk Bezk,n (u) K=0 K=0 K=0 n n n

Slide 49:

The entire curve lies within the convex hull of the control points.
This property ensures that no erratic* oscillations
Transforming a Bézier curve point by point is equivalent to first transforming the control points and then redrawing the curve
Note: * erratic irregular or unpredictable

Bezier surface :

Bezier surface 2 sets of orthogonal Bezier curves can be used to design an object surface by specifying by an input mesh of control points
Bezier surfaces have the same properties as Bezier curves
For each surface patch, we can select a mesh of control points in the xy plane.
Patches can be pieced together.

B-Spline curves and Surfaces :

B-Spline curves and Surfaces B-splines have 2 advantages over Bezier curves.
The degree of B-Spline polynomial can be set independently of the no: of ctrl points.
B-splines allow local ctrl over the shape of a spline curve or surface
Disadvantage is
B-splines are more complex than Bezier splines.

3 general classifications :

3 general classifications Uniform B-spline curve
When spacing btw knots is constant
Eg: {1,2,3,4,5} or { -0.5,0.0,0.5}
Open-uniform B-spline curve (or simply Open B-spline)
The knot spacing is uniform except at the ends where knot values are repeated d times.
Eg: {0,0,1,2,3,3} here d=2 and n=3
{1,1,1,2,3,4,5,5,5}
Non-uniform B-spline curve
We can specify any values and intervals for the knot vector
Can choose multiple internal knot values and unequal spacing btw the knot values
Eg: {0,1,2,3,3,4} or { 0,2,2,3,3,6}
Nonuniform B-splines provide increased flexibility in controlling a curve shape.

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